Sinusoidal wave
A sine wave or sinusoid is a mathematical curve that describes a smooth repetitive oscillation. It is named after the function sine, of which it is the graph. It occurs often in pure and applied mathematics, as well as physics, engineering, signal processing and many other fields. Its most basic form as a function of time (t) is:
where:
The sine wave is important in physics because it retains its wave shape when added to another sine wave of the same frequency and arbitrary phase and magnitude. It is the only periodic waveform that has this property. This property leads to its importance in Fourier analysis and makes it acoustically unique.
In general, the function may also have:
which is
The wavenumber is related to the angular frequency by:.
where λ (Lambda) is the wavelength, f is the frequency, and v is the linear speed.
This equation gives a sine wave for a single dimension; thus the generalized equation given above gives the displacement of the wave at a position x at time t along a single line. This could, for example, be considered the value of a wave along a wire.
In two or three spatial dimensions, the same equation describes a travelling plane wave if position x and wavenumber k are interpreted as vectors, and their product as a dot product. For more complex waves such as the height of a water wave in a pond after a stone has been dropped in, more complex equations are needed.
This wave pattern occurs often in nature, including wind waves, sound waves, and light waves.
A cosine wave is said to be "sinusoidal", because which is also a sine wave with a phase-shift of π/2 radians. Because of this "head start", it is often said that the cosine function leads the sine function or the sine lags the cosine.
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